on the almost sure convergence for dependent random vectors in hilbert spaces

Acta Math. Hungar., 139 (3) (2013), 276–285DOI: 10.1007/s10474-012-0275-7

First published online October 11, 2012

ON THE ALMOST SURE CONVERGENCE FORDEPENDENT RANDOM VECTORS IN HILBERT

SPACES

L. V. THANH∗

Department of Mathematics, Vinh University, Nghe An 42118, Vietname-mail: [email protected]

(Received March 9, 2012; accepted May 31, 2012)

Abstract. This work develops almost sure convergence of negatively asso-ciated random vectors in Hilbert spaces. Extensions of a result in [4] are given.Illustrative examples are provided.

1. Introduction

Joag-Dev and Proschan [3] introduced the concept of negative associa-tion of random variables. A collection {X1, . . . , Xn} of random variables issaid to be negatively associated if for any disjoint subsets A, B of {1, . . . , n}and any real coordinatewise nondecreasing functions f on R

|A| and g on R|B|,

(1.1) Cov(f(Xk, k ∈ A), g(Xk, k ∈ B)

)� 0

whenever the covariance exists, where |A| denotes the cardinality of A.The concept of negative association can be extended to R

d-valued ran-dom vectors as follows. A collection {X1, . . . , Xn} of R

d-valued randomvectors is said to be negatively associated if for any disjoint subsets A, B of{1, . . . , n} and any real coordinatewise nondecreasing functions f on R

|A|d

and g on R|B|d,

(1.2) Cov(f(Xk, k ∈ A), g(Xk, k ∈ B)

)� 0

whenever the covariance exists.

∗ This work was supported by the Vietnam’s National Foundation for Science and TechnologyDevelopment (NAFOSTED).

Key words and phrases: almost sure convergence, negatively associated random vector,Hilbert space.

Mathematics Subject Classification: 60F15, 60B11, 60B12.

0236-5294/$20.00 c© 2012 Akademiai Kiado, Budapest, Hungary

mailto:[email protected]

DEPENDENT RANDOM VECTORS IN HILBERT SPACES 277

An infinite family of Rd-valued random vectors is negatively associated

if every finite subfamily is negatively associated.This concept was extended in [4] to random vectors with values in

real separable Hilbert spaces. Let H be a real separable Hilbert spacewith orthonormal basis {ej , j ∈ B} and inner product 〈 ·, · 〉. A sequence{Xn, n � 1} of random vectors with values in H is said to be negativelyassociated if for any d � 1, the sequence of R

d-valued random vectors{(

〈Xi, e1〉, . . . , 〈Xi, ed〉), i � 1} is negatively associated.

A sequence {Xn, n � 1} of random vectors with values in H is saidto be blockwise negatively associated if for any k � 0, the random vectors{Xi, 2k � i < 2k+1

}are negatively associated. For the concept of blockwise

m-dependent random variables and the almost sure convergence for block-wise m-dependent random variables, we refer to Moricz [7], Moricz et al. [8],Stadtmuller and Thanh [10], Thanh [11,12] for references.

In [4], the authors established the Kolmogorov strong law of large num-bers for sequences of H-valued negatively associated random vectors by the“Khintchine–Kolmogorov convergence theorem and the Kronerker lemmaapproach”. In this paper, the result in [4] is extended to more general set-ting by another approach. We also establish the Marcinkiewicz–Zygmundtype strong law for H-valued blockwise negatively associated identically dis-tributed random vectors. An example is given to show that in our setting,the method in [4] cannot apply anymore.

We would like to mention that the strong convergence problem for se-quences of negatively associated and/or positively associated random vari-ables was also recently investigated in [1,2,13,14].

Throughout this paper, H will denote a real separable Hilbert space withorthonormal basis {ej , j ∈ B} and inner product 〈 ·, · 〉. The symbol C de-notes a generic constant (0 < C < ∞) which is not necessarily the same ineach appearance, and log denotes the logarithm to base 2.

2. Almost sure convergence: non-identically distributed cases

The following lemma establishes the maximal inequality for negativelyassociated random vectors in H . See [4] for a proof.

Lemma 2.1. Let {Xn, n � 1} be a sequence of mean 0 negatively asso-ciated random vectors in H . Then for all n � 1,

(2.1) E

⎛

⎝ max1�k�n

∥∥∥∥

k∑

i=1

Xi

∥∥∥∥

2⎞

⎠ �n∑

i=1

E‖Xi‖2.

The following theorem is the main result in this section. This theoremextends a result in [4] to a more general setting by another approach. Notealso that in our setting, the method in [4] does not apply anymore.

Acta Mathematica Hungarica 139, 2013

278 L. V. THANH

Theorem 2.2. Let {Xn, n � 1} be a sequence of mean 0 blockwise neg-ative associated random vectors in H . Let {bn, n � 1} be a nondecreasingsequence of positive constants satisfying

(2.2) infn�0

b2n+1

b2n

> 1 and supn�0

b2n+1

b2n

< ∞.

If

(2.3)∞∑

n=1

∑

j∈B

E∣∣ 〈Xn, ej 〉

∣∣ rn

brnn

< ∞,

where 1 � rn � 2, n � 1, then

(2.4)∑n

i=1 Xi

bn→ 0 a.s. as n → ∞.

Remark 2.3. Condition (2.3) cannot be replaced by the weaker condi-

tion∑

j∈BE| 〈Xn,ej 〉 |rn

brnn

= o(1), see Example 2.6.

Proof of Theorem 2.2. Note that the first half of (2.2) ensures thatlimn→∞ bn = ∞. For n � 1 and j ∈ B,

E(〈Xn, ej 〉I(∣∣ 〈Xn, ej 〉

∣∣ � bn))

2

b2n

+ P(∣∣ 〈Xn, ej 〉

∣∣ > bn)(2.5)

= 2b−2n

∫ bn

0xP(

∣∣ 〈Xn, ej 〉

∣∣ > x) dx

� 2b−rn

n

∫ bn

0xrn −1P(

∣∣ 〈Xn, ej 〉

∣∣ > x) dx � 2

E∣∣ 〈Xn, ej 〉

∣∣ rn

brnn

.

It thus follows from (2.3) that

∞∑

n=1

∑

j∈B

⎡

⎣E(〈Xn, ej 〉I(

∣∣ 〈Xn, ej 〉

∣∣ � bn))

2

b2n

+ P(∣∣ 〈Xn, ej 〉

∣∣ > bn)

⎤

⎦ < ∞.

(2.6)

For n � 1, j ∈ B, set

Y (j)n = −bnI(〈Xn, ej 〉 < −bn) + 〈Xn, ej 〉I(

∣∣ 〈Xn, ej 〉

∣∣ � bn)

+ bnI(

〈Xn, ej 〉 > bn), Yn =

∑

j∈B

Y (j)n ej .



It is well known that for all j ∈ B, {Y(j)n − EY

(j)n , n � 1} is a sequence

of blockwise negatively associated random variables, so {Yn − EYn, n � 1}is a sequence of blockwise negatively associated random vectors. By thedefinition of Yn,

∞∑

n=1

E‖Yn − EYn‖2

b2n

=∞∑

n=1

1b2n

∑

j∈B

E(Y (j)n − EY (j)

n )2 �

∞∑

n=1

1b2n

∑

j∈B

E(Y (j)n )

2

(2.7)

� 3∞∑

n=1

∑

j∈B

⎛

⎝P(∣∣ 〈Xn, ej 〉

∣∣ > bn) +

E(〈Xn, ej 〉I(∣∣ 〈Xn, ej 〉

∣∣ � bn))

2

b2n

⎞

⎠ .

Combining (2.6) and (2.7), we get

(2.8)∞∑

n=1

E‖Yn − EYn‖2

b2n

< ∞.

Set

Tk =1

b2k+1 − b2k

maxj<2k+1

∥∥∥∥

j∑

i=2k

(Yi − EYi)∥∥∥∥, k � 0.

For k � 0,

ET 2k � CE

⎛

⎝ 1b2k+1

maxj<2k+1

∥∥∥∥

j∑

i=2k

(Yi − EYi)∥∥∥∥

⎞

⎠

2

(2.9)

� C

b22k+1

2k+1−1∑

i=2k

E‖Yi − EYi‖2 � C2k+1−1∑

i=2k

E‖Yi − EYi‖2

b2i

by the first half of (2.2) and by Lemma 2.1. It thus follows from (2.8) that∑∞k=0 ET 2

k < ∞ and so by the Borel–Cantelli lemma,

(2.10) limk→∞

Tk = 0 a.s.

Assume that 2k � n < 2k+1,

∥∥∥∥

∑ni=1(Yi − EYi)

bn

∥∥∥∥ � 1

b2k

k∑

j=0

maxl<2j+1

∥∥∥∥

l∑

i=2j

(Yi − EYi)∥∥∥∥(2.11)


280 L. V. THANH

=b2k+1

b2k

k∑

j=0

b2j+1 − b2j

b2k+1 Tj � Ck∑

j=0

b2j+1 − b2j

b2k+1 Tj

by the second half of (2.2). It follows from (2.10), (2.11) and the Toeplitzlemma that

(2.12)∑n

i=1(Yi − EYi)bn

→ 0 a.s. as n → ∞.

Since Xn =∑

j∈B 〈Xn, ej 〉ej for all n � 1,

∞∑

n=1

P {Xn �= Yn} =∞∑

n=1

P{

‖Xn − Yn‖2 > 0}

(2.13)

=∞∑

n=1

P

{ ∑

j∈B

(〈Xn − Yn, ej 〉

)2> 0

}=

∞∑

n=1

P

{ ∑

j∈B

(〈Xn, ej 〉 − Y (j)n )

2> 0

}

�∞∑

n=1

∑

j∈B

P

{

(〈Xn, ej 〉 − Y (j)n )

2> 0

}�

∞∑

n=1

∑

j∈B

P {∣∣ 〈Xn, ej 〉

∣∣ > bn} < ∞

by (2.6). From (2.12) and (2.13), to obtain (2.4), it suffices to show that

(2.14)∑n

i=1 EYi

bn→ 0 as n → ∞.

Since EXn = 0 for all n � 1, E〈Xn, ej 〉 = 0 for all n � 1, j ∈ B. It followsthat

∞∑

n=1

‖EYn‖bn

�∞∑

n=1

1bn

∑

j∈B

‖EY (j)n ej ‖ =

∞∑

n=1

1bn

∑

j∈B

|EY (j)n |(2.15)

�∞∑

n=1

∑

j∈B

[P(

∣∣ 〈Xn, ej 〉

∣∣ > bn) +

1bn

∣∣∣E(〈Xn, ej 〉I(

∣∣ 〈Xn, ej 〉

∣∣ � bn))

∣∣∣]

=∞∑

n=1

∑

j∈B

[P(

∣∣ 〈Xn, ej 〉

∣∣ > bn) +

1bn

∣∣∣E(〈Xn, ej 〉I(

∣∣ 〈Xn, ej 〉

∣∣ > bn))

∣∣∣]

�∞∑

n=1

∑

j∈B

[P(

∣∣ 〈Xn, ej 〉

∣∣ > bn) +

1bn

E(∣∣ 〈Xn, ej 〉

∣∣I(

∣∣ 〈Xn, ej 〉

∣∣ > bn)|

)]



=∞∑

n=1

∑

j∈B

[2P(

∣∣ 〈Xn, ej 〉

∣∣ > bn) +

1bn

∫ ∞

bn

P(∣∣ 〈Xn, ej 〉

∣∣ > t) dt

]

�∞∑

n=1

∑

j∈B

[2P(

∣∣ 〈Xn, ej 〉

∣∣ > bn) +

1brnn

∫ ∞

bn

trn −1P(∣∣ 〈Xn, ej 〉

∣∣ > t) dt

]

�∞∑

n=1

∑

j∈B

[

2P(∣∣ 〈Xn, ej 〉

∣∣ > bn) +

E∣∣ 〈Xn, ej 〉

∣∣ rn

brnn

]

< ∞

by (2.3) and (2.6). In view of (2.15) and the Kronecker lemma, (2.14) fol-lows. �

Remark 2.4. (i) If rn ≡ 2, then

∞∑

n=1

∑

j∈B

E∣∣ 〈Xn, ej 〉

∣∣ rn

brnn

=∞∑

n=1

∑

j∈B

E〈Xn, ej 〉2

b2n

=∞∑

n=1

E‖Xn‖2

b2n

.

Therefore, Theorem 2.2 is an extension of the strong law of large numbersof Ko et al. [4] to a more general setting.

(ii) If the cardinality |B| of B is finite, then (2.3) is equivalent to

(2.16)∞∑

n=1

E‖Xn‖rn

brnn

< ∞.

Therefore, Theorem 2.2 is an extension of the classical Loeve strong law oflarge numbers to the blockwise negative association setting.

The following example is a modification of an example of Moricz [7] (seealso in Rosalsky and Thanh [9]) and it shows apropos of Theorem 2.2 thatunder its hypotheses the series

∑∞i=1 Xi/bi can diverge a.s. Consequently,

the conclusion (2.4) cannot in general be reached through the well-knownKronecker lemma approach for proving strong laws of large numbers whichwas used in [4,6].

Example 2.5. Let the underlying Hilbert space be the real line andlet {bn, n � 1} be as in Theorem 2.2. Let {Yn, n � 1} be a sequence ofmean 0 negatively associated random variables such that P {Y1 �= 0} = 1 and∑∞

n=1 EY 2n < ∞. For m � 0, set

Xn =

⎧⎪⎨

⎪⎩

b2mY1

m + 1, n = 2m

bnYn, 2m < n < 2m+1.


282 L. V. THANH

Then {Xn, n � 1} is a blockwise negatively associated sequence. Now

∞∑

n=1

EX2n

b2n

= EY 21

∞∑

m=0

1(m + 1)2

+∑

n�3, log n�∈N

EY 2n < ∞

and so (2.4) holds by Theorem 2.2. However, by Theorem 3 in [6],∑

3�i�n, log i �∈N

Yi converges a.s. as n → ∞

and

∑

1�i�n, log i∈N

Yi

1 + log i= Y1

[log n]∑

m=0

11 + m

diverges a.s. as n → ∞,

where [x] is the integer part of the real number x. Consequently, for n � 3,

n∑

i=1

Xi

bi=

∑

3�i�n, log i �∈N

Yi +∑

1�i�n, log i∈N

Yi

1 + log idiverges a.s. as n → ∞.

The second example shows that in Theorem 2.2, we cannot replace (2.3)

by the weaker condition∑

j∈BE| 〈Xn,ej 〉 |rn

brnn

= o(1).

Example 2.6. Let {bn, n � 1} be a sequence of positive constants sat-isfying (2.2) and such that

{b2n/(1 + log n), n � 1

}is nondecreasing. Let

{Yn, n � 1} be a sequence of mean 0 negatively associated random vectorsin H where

P{

‖Yn‖ = b2n−1}

=1

1 + log n, P {Yn = 0} = 1 − 1

1 + log n, n � 1.

Let Xn = Yn−2m+1, 2m � n < 2m+1, m � 0 and let rn ≡ 2. Then {Xn,n � 1} is a sequence of mean 0 blockwise negatively associated random vec-tors. Now X2m+1−1 = Y2m , m � 0 and so {X2m+1−1, m � 0} is a sequence ofmean 0 negatively associated random vectors with

P {∥∥X2m+1−1

∥∥ = b2m+1−1} =

1m + 1

, P {X2m+1−1 = 0} =m

m + 1, m � 0.

We see that∞∑

n=1

∑

j∈B

E∣∣ 〈Xn, ej 〉

∣∣ rn

brnn

=∞∑

n=1

E‖Xn‖2

b2n



�∞∑

m=0

E∥∥X2m+1−1

∥∥2

b22m+1−1

=∞∑

m=0

b22m+1−1

(m + 1)b22m+1−1

=∞∑

m=0

1m + 1

= ∞

and so (2.3) fails. We also have

(2.17)∞∑

m=0

P {∥∥X2m+1−1

∥∥ = b2m+1−1} =

∞∑

m=0

1m + 1

= ∞.

Hence, by the generalized Borel–Cantelli lemma (see, e.g., Kochen andStone [5]), (2.17) yields

P {∥∥X2m+1−1

∥∥ = b2m+1−1 i.o. (m)} = 1.

Thus,

1 = lim supm→∞

∥∥X2m+1−1

∥∥

b2m+1−1� lim sup

n→∞

‖Xn‖bn

� lim supn→∞

‖ ∑ni=1 Xi‖bn

+ lim supn→∞

‖ ∑n−1i=1 Vi‖bn−1

a.s.

implying

lim supn→∞

‖ ∑ni=1 Vi‖bn

� 12

a.s.

Thus (2.4) fails. We also note that for all large n, writing 2m � n < 2m+1,

∑

j∈B

E∣∣ 〈Xn, ej 〉

∣∣ rn

brnn

=E‖Xn‖2

b2n

=E‖Yn−2m+1‖2

b2n

=b22(n−2m+1)−1

b2n

(1 + log (n − 2m + 1)

) �b22m+1−1

b22m(1 + log 2m)

� C

m + 1→ 0 as m → ∞

by the second half of (2.2).

3. The identically distributed case

In this section, we establish the Marcinkiewicz–Zygmund type strong lawof large numbers for sequences of blockwise negatively associated randomvectors. Our main result in this section is the following theorem.


284 L. V. THANH

Theorem 3.1. Let {Xn, n � 1} be a sequence of mean 0 blockwise neg-atively associated random vectors in H . If {Xn, n � 1} are identically dis-tributed with

(3.1)∑

j∈B

E∣∣ 〈X1, ej 〉

∣∣ p

< ∞ for some 1 � p < 2,

then

(3.2)∑n

i=1 Xi

n1/p→ 0 a.s. as n → ∞.

Proof. For n � 1, j ∈ B, set

Y (j)n = −n1/pI(〈Xn, ej 〉 < −n1/p) + 〈Xn, ej 〉I(

∣∣ 〈Xn, ej 〉

∣∣ � n1/p)

+ n1/pI(

〈Xn, ej 〉 > n1/p) , Yn =∑

j∈B

Y (j)n ej .

By (3.1) and a standard calculation, we have

(3.3)∞∑

n=1

E‖Yn − EYn‖2

n2/p< ∞.

By the same way as we did in the proof of Theorem 2.2, we see that (2.12)still holds. The rest of the proof follows by a routine argument. �

Remark 3.2. Similar to Remark 2.4 (ii), if the cardinality |B| of Bis finite, then (3.1) is equivalent to E‖X1‖p < ∞. Therefore, Theorem 3.1is an extension of the classical Marcinkiewicz–Zygmund strong law of largenumbers to the blockwise negative association setting.

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[10] U. Stadtmuller and L. V. Thanh, On the strong limit theorems for double arrays ofblockwise M -dependent random variables, Acta Math. Sin. (Engl. Ser.), 27(2011), 1923–1934.

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on the almost sure convergence for dependent random vectors in hilbert spaces

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